Is the inner product a bilinear function in vector spaces?

[quasar_4]

Hello all,

I have two questions that are fairly general, but slightly hazy to me still.

1) Can we consider the inner product to be a bilinear function, or not? I would like to think of it as a mapping from an ordered pair of vectors of some vector space V (i.e. VxV) to the field (F), and I know by definition the inner product is conjugate linear as a function of it's first entry (or second, depending on which text you use). But isn't the inner product also linear as a function of either entry whenever the other is held fixed, making it bilinear?

2) Can every mapping from a finite dimensional vector space V to it's field be considered an inner product of something? What about the infinite dimensional case?

 

[matt grime]

1) Yes, obviously: what is a bilinear map?

2) No, equally obviously. If you allow mapping to mean linear functional then you need the Reitz representation theorem. But since very few vector spaces have inner products the answer is still 'NO', for equally obvious reasons.